Ex 9.3

Chapter 9 Class 12 Differential Equations
Serial order wise

Transcript

Ex 9.3, 1 For each of the differential equations in Exercises 1 to 10, find the general solution : ππ¦/ππ₯=(1 β cosβ‘π₯)/(1 + cosβ‘π₯ ) ππ¦/ππ₯ = (1 β cosβ‘π₯)/(1 + cosβ‘π₯ ) We know that cos 2x = 2cos2 x β 1 Putting x = π₯/2 cos 2π₯/2 = 2 cos2 π₯/2 β 1 cos x = 2 cos2 π₯/2 β 1 1 + cos x = 2cos2 π₯/2 We know cos 2x = 1 β 2sin2 x Putting x = π₯/2 cos2 2π₯/2 = 1 β 2 sin2 π₯/2 cos x = 1 β 2sin2 π₯/2 1 β cos x = 2sin2 π₯/2 Putting values in equation ππ/ππ = (π γγπ¬π’π§γ^π γβ‘γπ/πγ )/(π γγππ¨π¬γ^π γβ‘γπ/πγ ) ππ¦ = (γsin^2 γβ‘γπ₯/2γ )/cos^(2 )β‘γπ₯/2γ ππ₯ ππ¦ = tan2 π₯/2 ππ₯ Putting tan2 π₯/2 = sec2 π₯/2 β 1 ππ = ("sec2 " π/π " β 1" )ππ Integrating both sides We know that tan2 x + 1 = sec2 x tan2 x = sec2 x β 1 Putting x = π₯/2 tan2 π/π = sec2 π/π β 1 β«1βππ = β«1β(ππππ π/πβπ) ππ y = β«1βγπ ππ2 π₯/2 ππ₯β β«1βππ₯γ y = π/(π/π) tan π/π β x + C y = 2 tan π₯/2 β x + c y = 2 tan π₯/2 β x + C y = π tan π/π β x + C is the general solution